- Write the first terms of the sequences whose nth terms is an = n(n + 2)
Solution:
an = n(n + 2)
Substituting n = 1, 2, 3, 4 and 5 we obtain
a1 = 1(1+2) = 3
a2 = 2(2+2) = 8
a3 = 3(3+2) = 15
a4 = 4(4+2) = 24
a5 = 5(5+2) = 35
Therefore, the required terms are 3, 8, 15, 24 and 35.
- Write the first five terms of the sequences whose nth terms is an = n/(n+1)
Solution:
an = n/(n+1)
Substituting n = 1, 2, 3, 4, 5 obtain
a1 = 1/1+1 = 1/2 ,
a2 = 2/2+1 = 2/3 ,
a3 = 3/3+1 = 3/4 ,
a4 = 4/4+1 = 4/5 ,
a5 = 5/5+1 = 5/6 ,
Therefore, the required terms are 1/2, 2/3 , 3/4 , 4/5 and 5/6
- Write the first five terms of the sequences whose nth term is an = 2n
Solution:
an = 2n
Substituting n = 1, 2, 3, 4, 5 we obtain
a1 = 21 = 2
a2 = 22 = 4
a3 = 23 = 8
a4 = 24 = 16
a5 = 25 = 32
Therefore the required terms are 2, 4, 8, 16 and 32
- Write the first five terms of the sequences whose nth terms is an = (2n-3)/6
Solution:
Substituting n = 1, 2, 3 , 4, 5 we obtain
a1 = (2×1-3)/6 = -1/6 ,
a2 = (2×2-3)/6 = 1/6 ,
a3 = (2×3-3)/6 = 3/6 = 1/2 ,
a4 = (2×4-3)/6 = 5/6 ,
a5 = (2×5-3)/6 = 7/6 ,
Therefore, the required terms are -1/6 , 1/6 , 1/2 , 5/6 and 7/6 .
- Write the first five terms of the sequences whose nth terms is an = (-1)n-15n+1
Solution:
Substituting n = 1, 2, 3, 4, 5 we obtain
a1 = (-1)1-1 51+1 = 52 = 25
a2 = (-1)2-1 52+1 = -53 = -125
a3 = (-1)3-1 53+1 = 54 = 625
a4 = (-1)4-1 54+1 = -55 = -3125
a5 = (-1)5-1 55+1 = 56 = 15625
Therefore, the required terms are 25, -125, 625, -3125 and 15625.
- Write the first five terms of the sequences whose nth term is an = n[(n2+5)/4]
Solution:
Substituting n = 1, 2, 3, 4, 5 we obtain
a1 = 1. [(12+5)/4] = 6/4 = 3/2,
a2 = 2. [(22+5)/4] = 2. 9/4 = 9/2,
a3 = 3. [(32+5)/4] = 3. 14/4 = 21/2,
a4 = 4. [(42+5)/4] = 21,
a5 = 5. [(52+5)/4] = 5. 30/4 = 75/2,
Therefore, the required terms are 3/2, 9/2, 21/2, 21 and 75/2.
- Find the 17th in the following sequences whose nth term is an = 4n – 3 , a17, a24
Solution:
Substituting n = 17 we obtain
a17 = 4(17) – 3 = 68 – 3 = 65
Substituting n = 24 we obtain
a24 = 4(24) – 3 = 96 – 3 = 93
- Find the 7th term in the following sequences whose nth terms is an = [(n2)/2n]; a7
Solution:
Substituting n = 7 we obtain
a7 = (72)/(27) = 49/128
- Find the 9th term in the following sequences whose nth term is an = (-1)n-1n3; a9
Solution:
Substituting n = 9 we obtain
a9 = (-1)9-1(9)3 = (9)3 = 729
- Find the 20th term in the following sequences whose nth terms is an = [{n(n+2)}/(n+3)]; a20
Solution:
Substituting n = 20, we obtain
a20 = [{20(20+2)}/(20+3)] = 20(18)/23 = 360/23
- Write the first five terms of the following sequences and obtain the corresponding series:
a1 = 3, an = 3an-1 + 2, for all n > 1
Solution:
a1 = 3, an = 3an-1 + 2 for all n>1
a2 = 3a1 + 2 = 3(3) + 2 = 11
a3 = 3a2 + 2 = 3(11) + 2 = 35
a4 = 3a3 + 2 = 3(35) + 2 = 107
a5 = 3a4 + 2 = 3(107) + 2 = 323
Hence the first five terms of the sequences are 3, 11, 35, 107 and 323.
The corresponding series 3 + 11+ 35 + 107 + 323 + …
- Write the first five terms of the following sequence and obtain the corresponding series:
a1 = -1, an = [(an-1)/n], n ≥ 2
Solution:
a1 = -1, an = [(an-1)/n], n ≥ 2
a2 = (a1)/2 = -1/2,
a3 = (a2)/3 = -1/6,
a4 = (a3)/4 = -1/24,
a5 = (a4)/4 = -1/120,
Hence the first five terms of the sequences are -1, -1/2, -1/6, -1/24, -1/120,
The corresponding series is (-1)+( -1/2)+( -1/6)+( -1/24)+( -1/120)+…
- Write the first five terms of the following sequence and obtain the corresponding series:
a1 = a2 = 2, an = an-1 -1, n > 2
Solution:
a1 = a2 = 2, an = an-1 -1, n > 2
a3 = a2 – 1 = 2 – 1 = 1
a4 = a3 – 1 = 1 – 1 = 0
a5 = a4 – 1 = 0 – 1 = -1
Hence, the first five terms of the sequence are 2, 2, 1, 0 and -1 and the corresponding series is 2 + 2 + 1 + 0 + (-1) + …
- The Fibonacci sequence is defined by 1 = a1= a2 and an = an-1 + an-2, n>2
Find [(an+1)/(an)], for n = 1, 2, 3, 4, 5
Solution:
1 = a1= a2
an = an-1 + an-2, n>2
a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 2 + 1 = 3
a5 = a4 + a3 = 3 + 2 = 5
a6 = a5 + a4 = 5 + 3 = 8
For n = 1, [(an+1)/(an)] = [(a2)/(a1)] = 1/1 = 1
For n = 2, [(an+1)/(an)] = [(a3)/(a2)] = 2/1 = 2
For n = 3, [(an+1)/(an)] = [(a4)/(a3)] = 3/2
For n = 4, [(an+1)/(an)] = [(a5)/(a4)] = 5/3
For n = 5, [(an+1)/(an)] = [(a6)/(a5)] = 8/5
